Complexity, Geometry, and Self-organization in Pilgrimage Systems:

I. THE NON-EQUILIBRIUM UNIVERSE

When two different liquids are placed in the same vessel they will diffuse into a homogeneous mixture, thereby achieving an equilibrium state with the greatest disorder and maximum entropy. Such behavior is the prediction of the Second Law of Thermodyn amics. At some time far into the future all configurations of matter and energy in our universe will be similarly homogenized and will approach the condition of thermodynamic equilibrium. Stars will cool and eventually temperature differences between them and their surroundings will vanish. Depending upon whether or not the universe collapses, even black holes and neutron stars may dissipate their concentrated mass and fade into a homogeneous background. In its most perfect form, thermodynamic equilibrium involves a detailed balancing of all processes, such that each process is precisely balanced by its inverse. In such a balance no information, inhomogeneity, or complexity can persist. The state of thermodynamic equilibrium is frozen in time, remembering no history, containing no information, and incapable of sustained change.

Today portions of our universe depart significantly from equilibrium. The universe of stars, galaxies, and clusters of galaxies is highly inhomogeneous: hot, bright, dense stars are surrounded by cold dark vacuous space. The expansion of the boundaries of the universe prevents stagnation, insures the universe's continuing departure from equilibrium, and allows the formation of complex low entropy structures. Within the non-equilibrium biosphere of planet earth there are dense concentrations of informat ion combined with complex and unlikely behavior.